where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translationa

where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translational degrees of freedom of each particle and the Boltzmann constant, converting units between the microscopic model and the macroscopic temperature. This formalism is the basic assumption that directly yields the ideal gas law, and it shows that for the ideal gas, the internal energy consists only of its thermal energy:

» where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translationa
» where k is the Boltzmann constant. The total thermal energy of a sample of matter or a thermodynamic system is consequently the average sum of the kinetic energies of all particles in the system. Thus, for a system of N particles its thermal energy is[3]
» where k is the Boltzmann constant. The total thermal energy of a sample of matter or a thermodynamic system is consequently the average sum of the kinetic energies of all particles in the system. Thus, for a system of N particles its thermal energy is[3]
» where m is the particle's mass and v is its velocity. The thermal energy of the gas sample consisting of N atoms is given by the sum of these energies, assuming no losses to the container or the environment:
» The term thermal energy is also often used as a property of single particles to designate the kinetic energy of the particles. An example is the description of thermal neutrons having a certain thermal energy, which means that the kinetic energy of the pa